Processes with the aforedescribed process features are generally known as laser welding processes. The thus observed optical and acoustical signals are used for monitoring and control. Such signals arise from the workpiece surface or from above the workpiece surface. It is thus possible, for example, to detect a weld breakthrough only in isolated parameter regions. It is also generally known to observe the underside of a workpiece to detect a weld breakthrough. The workpiece is, however, during industrial finishing processes generally only accessible with difficulty at the underside. It is therefore necessary to detect the penetration depth in the workpiece from the upper side, i.e. from the side to which the laser radiation is fed.